Properties

Label 2-5610-1.1-c1-0-83
Degree $2$
Conductor $5610$
Sign $-1$
Analytic cond. $44.7960$
Root an. cond. $6.69298$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 3.12·7-s − 8-s + 9-s + 10-s − 11-s − 12-s + 5.12·13-s − 3.12·14-s + 15-s + 16-s − 17-s − 18-s + 3.12·19-s − 20-s − 3.12·21-s + 22-s − 3.12·23-s + 24-s + 25-s − 5.12·26-s − 27-s + 3.12·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.18·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.42·13-s − 0.834·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.716·19-s − 0.223·20-s − 0.681·21-s + 0.213·22-s − 0.651·23-s + 0.204·24-s + 0.200·25-s − 1.00·26-s − 0.192·27-s + 0.590·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(44.7960\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 - 3.12T + 7T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 7.12T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 9.12T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 4.87T + 71T^{2} \)
73 \( 1 + 1.12T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.69282559511759486298463732786, −7.42742845960879789994159223270, −6.32504887213968881735474383656, −5.72001551675743498886300221038, −4.98629576348545896343190146445, −4.06853825893635969694119455988, −3.31958827164474444638377309570, −1.92339056725753052652835142211, −1.30233501805382164955410724687, 0, 1.30233501805382164955410724687, 1.92339056725753052652835142211, 3.31958827164474444638377309570, 4.06853825893635969694119455988, 4.98629576348545896343190146445, 5.72001551675743498886300221038, 6.32504887213968881735474383656, 7.42742845960879789994159223270, 7.69282559511759486298463732786

Graph of the $Z$-function along the critical line